EDIT: from a previous post of PJT
You have a couple of options. There is the Santo Domingo to Bavaro bus (809) 686-0637 that leaves Sto Domingo from Juan Sanchez Ramirez #31 near Maximo Gomez, express direct to Bavaro/Punta Cana. The bus leaves at 7 am and 4 pm, the cost is 170 Pesos. 7 am bus arrives Bavaro/Punta Cana around 11 am, 4 pm bus arrives around 8-9 pm
Your other option is to go to Parque Enriquillo in the Capital and catch an hourly express bus to Higuey for about 120 Pesos and then take a local mini-bus (gua-gua) about 35 Pesos to Bavaro/Punta Cana.
Note: If you are traveling with baggage be advised the gua-guas from Higuey do not have stowage. What you have with you will have to be placed in you lap or you pay for the seat to stow it. Plus, the gua-guas are crowded. Better bet is a taxi from Higuey to Bavaro/Punta Cana that will cost about $400 Pesos.
Note: The express buses to/from the Capitol are clean with a/c and movies.
Regards, PJT